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Dissertations, Theses, and Masters Projects Theses,
Dissertations, & Master Projects
2003
Multinuclear NMR studies of relaxor ferroelectrics Multinuclear
NMR studies of relaxor ferroelectrics
Donghua Zhou College of William & Mary - Arts &
Sciences
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Recommended Citation Recommended Citation Zhou, Donghua,
"Multinuclear NMR studies of relaxor ferroelectrics" (2003).
Dissertations, Theses, and Masters Projects. Paper 1539623422.
https://dx.doi.org/doi:10.21220/s2-xsb9-p144
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Multinuclear NMR Studies of Relaxor Ferroelectrics
A D issertation
Presented to
The Faculty of the D epartm ent of Physics
The College of W illiam & M ary in Virginia
In P artia l Fulfillment
Of the Requirem ents for the Degree of
Doctor of Philosophy
by
Donghua Zhou
2003
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APPROVAL SHEET
This dissertation is subm itted in partial fulfillment of
the requirem ents for the degree of
D octor of Philosophy
Approved, M arch 2003
Donghua Zhou
f a Os L . R o exAs o nGina L. Hoatson
Advisor
Henry Krakauer
enneth G. PetzineVr
Robert L. Void Department of Applied Science
Brian Holloway Department of Applied Science
ii
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To my parents and Lingjin.
iii
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Table of Contents
A ck n ow led gm en ts v iii
L ist o f T ab les ix
L ist o f F igures x ii
A b stra ct x iii
1 In tro d u ctio n 2
2 N M R S p ec tro sco p y 4
2.1 Precessing M agnetization and NM R Signal
............................................................ 6
2.2 Density M a t r i x
................................................................................................................
8
2.3 NM R S p ec tro m ete
r.........................................................................................................
11
2.4 Relaxation T im e s
.............................................................................................................
13
2.5 Recovery Times and Spin E c h o
...................................................................................
15
2.6 S u m m a r y
..........................................................................................................................
17
3 C h em ica l Shift In tera c tio n 18
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3.1 Chemical Shielding and Chemical S h i f t s
..................................................................
19
3.2 S tatic Spectra
.................................................................................................................
21
3.3 Magic-Angle Spinning S p e c tra
......................................................................................
23
3.4 Total Suppression of Spinning S id eb an d
s.................................................................
26
3.5 Phase A djusted Spinning Sidebands
........................................................................
28
3.6 S u m m a r y
...........................................................................................................................
34
4 E lectr ic Q u ad ru p ole In tera c tio n 36
4.1 Q uadrupolar Nuclei and Q uadrupolar In te ra c tio n
................................................ 36
4.2 P ertu rbation Expansion of Energy Levels and Transition
Frequencies . . . 47
4.3 Quadrupole Spectra of Polycrystalline M aterials
................................................. 49
4.4 The Quest to Remove A n iso tro p y
...............................................................................
57
4.5 P rototype Experim ent and Theory of M Q M A S
.................................................... 60
4.6 Pure Absorption L in e s h a p e
.........................................................................................
66
4.7 E xtraction of Param eters and Site P o p u la t io n s
.................................................... 74
4.8 Axes of D istributions
...................................................................................................
76
4.9 “Dream ” Transform ation
.............................................................................................
76
4.10 S u m m a r y
...........................................................................................................................
79
5 P ero v sk ite R elaxor F erroe lectr ics 81
5.1 F e rro e le c tr ic s
....................................................................................................................
81
5.2 Relaxor Ferroelectrics
...................................................................................................
82
5.3 Origins of Relaxor Behaviors
......................................................................................
85
5.4 NM R Studies on PM N-PSN Solid S o lu tio n s
........................................................... 90
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6 N io b iu m N M R S tu d ies o f P M N -P S N 94
6.1 High Field and Fast Speed MAS N M R
....................................................................
95
6.2 MAS NM R Results
.....................................................................................................
96
6.3 Spectral Assignments
..................................................................................................
100
6.4 Models of cation disorder
...........................................................................................
106
6.5 Higher Resolution Needed: 3QMAS N M R
............................................................
112
6.6 3QMAS Results
............................................................................................................
114
6.7 F it of 3QMAS Spectra
..............................................................................................
125
6.8 C o n c lu s io n s
......................................................................................................................
127
7 S cand ium N M R S tu d ies o f P M N -P S N 131
7.1 MAS
................................................................................................................................
133
7.2 3 Q M A S
.............................................................................................................................
136
7.3 Conclusion
.....................................................................................................
142
8 Lead N M R S tu d ies o f P M N -P S N 143
8.1 Experim ental Setting
..................................................................................................
145
8.2 Static and MAS spectra, and T\ M e a su re m e n ts
.................................................... 147
8.3 2D-PASS: Isotropic Chemical S h i f t
..........................................................................
149
8.4 M easurement of the shortest P b -0 bond l e n g t h
.................................................. 153
8.5 Lead displacement m o d e ls
...........................................................................................
155
8.6 2D-PASS: Anisotropic Chem ical Shift
....................................................................
158
8.7 P b -0 bonding en v iro n m en ts
.......................................................................................
160
8.8 Conclusion
......................................................................................................................
162
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9 C on clu sion 164
A R o ta tio n M atr ices 167
B C oh eren ce Transfer P a th w ay and P h a se C yclin g
169
B .l Traditional S c h e m e
.........................................................................................................
170
B.2 Simplification of Phase C y c l in g
...................................................................................
172
B.3 S u m m a r y
..........................................................................................................................
172
B ib liograp h y 175
vii
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ACKNOWLEDGMENTS
I would like to express my gratitude to my advisors Drs. G ina
L. Hoatson and Robert
L. Void for their guidance and patience. Their quality to pursue
perfection has deeply
influenced me. All the com m ittee members are thanked for
spending tim e reading and
commenting on my dissertation. Dr. Henry K rakauer and Dr.
Shiwei Zhang are thanked
for several helpful discussions. Bill Brouwer read m any
chapters and corrected my English.
I highly appreciate the generous help and friendship from
colleagues in the NM R lab: Dr.
D ariya (Dasha) M alyarenko, Yana Goddard, Dr. Jprgcn
Kristensen, Dr. Sixun Zheng, Bill
Brouwer, E lizabeth Slonaker, Jason Gammon, and Xin Zhao.
Many people outside the departm ent deserve my thanks. Dr. Peter
K. Davies in the
University of Pennsylvania m ade this work possible by providing
us well characterized re
laxor ferroelectric samples. It was with Drs. Dominique M assiot
and Franck Fayon in
CNRS, France, G ina and Bob sta rted this project during their
sabbatical (Septem ber 2000
to M arch 2001). Dr. M assiot generously allowed us to use his
spectral analyzing software
D M F IT and Dr. Fayon com m ented the 93Nb 3QMAS and 207Pb
2D-PASS m anuscripts.
Dr. Zhehong G an in NHMFL, Florida patiently assisted Gina and
Donghua in acquiring
93Nb and 45Sc spectra using the 19.6 Tesla spectrom eter.
My wife Lingjin took good care of me so th a t I was able to
focus on th is project. I
would like to thank her for her love, dedication, patience, and
support.
viii
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List of Tables
2.1 M agnetic properties
......................................................................................................
5
3.1 Delays between pulses for 2D-PASS s e q u e n c e
...................................................... 31
3.2 Improved phase cycling for 2 D -P A S S
.......................................................................
33
4.1 Quadrupole m o m en ts
......................................................................................................
44
4.2 Sternheim er f a c to rs
.........................................................................................................
45
4.3 Essential coefficients of MQMAS and S T M A S
...................................................... 64
4.4 NM R param eters for the three sites in Na2SC> 3
................................................... 74
6.1 Deconvolution param eters of MAS s p e c t r a
.............................................................
100
6.2 Random site prediction and experim ental r e s u l t s
............................................... 110
6.3 NM R param eters for the narrow peak and the two broad p e a
k s .................... 117
7.1 Decomposition param eters for 45Sc MAS s p e c t r a
............................................... 135
8.1 G aussian fit param eters for the isotropic p r o je c t io
n s ........................................ 153
8.2 Param eters of unique direction model
...................................................................
157
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List of Figures
2.1 C reation of B\ f ie ld
........................................................................................................
7
2.2 Free induction decay and s p e c t r u m
..........................................................................
8
2.3 An NM R sp e c tro m e te r
.................................................................................................
12
2.4 T\ m easurem ent using satu ration c o m b
...................................................................
14
2.5 Spin echo pulse sequence
..............................................................................................
16
3.1 S tatic and MAS 207Pb NM R spectra of PbS0 4
.................................................. 23
3.2 Exam ple of 2D-PASS s p e c t r u m
................................................................................
29
3.3 2D-PASS pulse se q u e n c e
..............................................................................................
30
3.4 2D-PASS 207P b NM R spectra of P b S 0 4
...............................................................
34
4.1 S tatic powder p a tte rn of first order quadrupole in te
rac tio n ............................. 50
4.2 S tatic and MAS powder patterns of second order quadrupole
interaction . . 52
4.3 23Na static and MAS spectra of sodium o x a la te
.................................................. 54
4.4 The singularities of second order quadrupole lineshapes
................................. 55
4.5 E xtracting NM R param eters from MAS sp e c tru m
.............................................. 57
4.6 A prototype two pulse MQMAS s e q u e n c e
............................................................ 61
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4.7 MQMAS spectral p rocessing
.......................................................................................
62
4.8 Lineshapes in 2D spectra
...........................................................................................
68
4.9 MQMAS pulse sequence w ith whole-echo d e te c t io n
........................................... 69
4.10 Am plitude m odulation w ith Z -filte r
..........................................................................
71
4.11 23Na spectra of Na2S0 3
..............................................................................................
73
4.12 F itting the three sites in N a2S03 3QMAS and MAS s p e c t
r a .......................... 75
4.13 “Dream transform ation” of 3QMAS s p e c t r a
......................................................... 78
5.1 Generic perovskite s tr u c tu r e
.......................................................................................
83
5.2 Tem perature dependence of dielectric constant
.................................................. 83
5.3 X-ray diffraction s p e c t r a
..............................................................................................
91
5.4 TEM images of P M N -P S N
...........................................................................................
92
5.5 Dielectric constants for P M N - P S N
..........................................................................
93
6.1 93Nb MAS spectra of P M N
.......................................................................................
97
6.2 The seven narrow p e a k s
..............................................................................................
98
6.3 Constrained deconvolution of PM N MAS s p e c t r a
.............................................. 99
6.4 Deconvolution of PM N-PSN MAS s p e c t r a
............................................................
101
6.5 Perovskite s tructu re and B -la ttic e
.............................................................................
102
6.6 The twenty-eight nBn c o n fig u ra tio n s
......................................................................
104
6.7 Com parison of MAS spectra of PM N and P y ro c h lo re
........................................ 105
6.8 Random site p re d ic tio n s
..............................................................................................
I l l
6.9 93Nb 3QMAS spectra for P M N - P S N
......................................................................
114
6.10 Lineshape fits for a slice of PM N 3QMAS sp e c tru m
........................................... 116
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6.11 93Nb quadrupole products for scandium-deficient nBn
configurations . . . . 118
6.12 Oxygen octahedral distortions of two configurations
........................................ 119
6.13 Isotropic chemical shifts and quadrupole products of the
narrow peaks . . . 120
6.14 Assignments according to 3QMAS
..........................................................................
123
6.15 F it the PM N 3QMAS s p e c tru m
................................................................................
127
7.1 45 Sc MAS s p e c tra
...........................................................................................................
133
7.2 Decomposition of 45Sc MAS s p e c t r a
......................................................................
134
7.3 45Sc 3QMAS s p e c t r a
.....................................................................................................
137
7.4 Schematic 3QMAS spectra of P M N -P S N
...............................................................
140
8.1 Static and MAS 207P b spectra
................................................................................
147
8.2 Lead relaxation tim e T \
..............................................................................................
148
8.3 207P b 2D-PASS spectrum of P M N
.........................................................................
149
8.4 Isotropic projections of 207P b 2D-PASS s p e c t r a
.................................................. 151
8.5 Tetragonal structure of P b T i0 3
................................................................................
154
8.6 Shell distribution model for P b d isp la c em e n ts
..................................................... 156
8.7 Unique direction model for P b d is p la c e m e n ts
..................................................... 158
8.8 Correlation between anisotropic and isotropic chemical s h i
f t s ....................... 159
8.9 The tem perature dependence of the
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ABSTRACT
M ultinuclear NM R of 93Nb, 45Sc, and 207P b has been carried
out to study the structure, disorder, and dynamics of a series of
im portant solid solutions: perovskite relaxor ferroelectric m
aterials (1 — x) Pb(M g1/ 3N b2/ 3)0 3 -x P b (S c 1//2N b1//2)03
(PM N-PSN).
First, im portant interactions (including chemical shielding
anisotropy and quadrupole interactions) and relevant NM R
techniques such as magic-angle spinning (MAS), triplequantum MAS
(3QMAS), and two-dimensional phase-adjusted spinning side-bands
(20- PASS) are introduced.
93Nb NM R investigations of the local s truc tu re and cation
order/d isorder are presented as a function of PSN concentration,
x. T he superb fidelity and accuracy of 3QMAS allows us to make
clear and consistent assignm ents of spectral intensities to the 28
possible nearest B-site neighbor (nBn) configurations, (Nms , N$c,
where each num ber rangesfrom 0 to 6 and their sum is 6 . For most
of the 28 possible nBn configurations, isotropic chemical shifts
and quadrupole product constants have been extracted from the data
. The seven configurations w ith only larger cations, M g2+ and
Sc3+ (and no N b5+) are assigned to the seven observed narrow
peaks, whose deconvoluted intensities facilitate quantitative
evaluation of, and differentiation between, different models of
B-site (chemical) disorder. The “completely random ” model is ruled
ou t and the “random site” model is shown to be in qualitative
agreement w ith the NM R experim ents. To obtain quantita tive
agreement w ith observed NM R intensities, the random site model is
slightly modified by including unlike-pair interaction
energies.
To date, 45Sc studies have not been as fruitful as 93Nb NMR
because the resolution is lower in the 45Sc spectra. The lower
resolution of 45Sc spectra is due to a smaller span of isotropic
chemical shift (40 ppm for 45Sc vs. 82 ppm for 93Nb) and to the
lack of a fortuitous mechanism th a t simplifies the 93Nb spectra;
for 93Nb the overlap of the isotropic chemical shifts of 6-Sc and
6-Nb configurations results in the alignment of all the 28
configurations along only seven quadrupole distribution axes.
Finally we present variable tem perature 207Pb static, MAS, and
2D-PASS NM R studies. Strong linear correlations between isotropic
and anisotropic chemical shifts show th a t P b -0 bonds vary from
more ionic to more covalent environments. D istributions of P b -0
bond lengthes are also quantitatively described. Such distributions
are used to examine two competing models of Pb displacements; the
shell model and the unique direction model. Only the la tte r model
is able to reproduce th e observed P b -0 distance
distribution.
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Multinuclear NMR Studies of Relaxor Ferroelectrics
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Chapter 1
Introduction
Nuclear M agnetic Resonance (NMR) is a powerful spectroscopic
technique th a t provides
information about the structure and dynamics of m atter on the
molecular level. After the
discovery of NM R in bulk m aterials in 1945 by Edw ard M.
Purcell, Henry C. Torrey, and
Robert V. Pound and independently by Felix Bloch, W illiam W.
Hansen, and M artin E.
Packard, NM R has been continuously undergoing trem endous
advances: from continuous
wave to pulse Fourier transform NMR, from one dimension to m
ultiple dimensions, from
one nucleus to several coupled nuclei, and from static to
spinning solid samples a t the so-
called magic angle. The range of its applications have been
extended beyond physics, to
chemistry, geology, biology, and medicine. Three Nobel prizes
have been awarded to NM R
works, testifying th a t NM R is still very vigrous after a half
century: the 1952 physics prize
was awarded jointly to Bloch and Purcell for their discovery of
NMR; the 1991 chemistry
prize was awarded to Richard R. E rnst for his contributions to
the development of the
m ethodology of high resolution NM R (especially F T NM R and
multi-dim ension N M R ); half
of the 2002 the chemistry prize was awarded to K urt W iithrich
for his development of NM R
spectroscopy for determ ining the three-dimensional structure of
biological macromolecules
in solution.
2
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CH APTER 1. INTRO DU CTION 3
In this work NM R is used to study relaxor ferroelectric
ceramics, whose im portant appli
cations in electronics and optics include, bu t are not lim ited
to, capacitors, nonvolatile mem
ories, medical ultrasound generators and receivers, high
frequency speakers, displacement
transducers, accelerometers, voltage transform ers, gas
ignitors, flash protection goggles, and
optical memories and displays. The local structures and
short-range order/d isorder of these
m aterials are crucial for their special properties. However,
X-ray diffraction is of lim ited
use since there is little or no long range order. NM R is an
powerful tool to study local
structures and disorder; this work proves th a t NM R is able to
shed valuable light on the
understanding of the microscopic origin of the properties of
relaxor ferroelectrics.
C hapter 2 introduces the basics of NM R spectroscopy. Chapter 3
focuses on chemical
shielding effects, which may be used to probe the surrounding
electronic wave function; tech
niques such as magic-angle spinning (MAS) and two-dimensional
phase-adjusted spinning
sidebands (2D-PASS) are introduced to achieve high resolution
and to obtain the chem
ical shielding (or shift) tensor. C hapter 4 focuses on the
quadrupole interaction, where
the nuclei are used to probe the local electric field gradients;
techniques such as MAS and
m ulti-quantum MAS (MQMAS) are described. C hapter 5 introduces
the properties of the
relaxor ferroelectrics, it also reviews the endeavors m ade to
understand these m aterials from
other fields such as X-ray and neutron diffraction, electron
microscopy, and com putational
modelling. C hapter 6 describes the 93Nb MAS and 3QMAS studies
of a series of relaxor
ferroelectrics, (1 — x)PbM g1/ 3Nb2/ 30 3 -a;PbSc1/ 2N b1/203
(PM N-PSN); local s tructu re and
quantitative inform ation on cation disorder is obtained.
Chapter 7 describes the 45 Sc MAS
and 3QMAS studies of PM N-PSN. Finally, chapter 8 describes the
study of local structures
by 207P b static, MAS, and 2D-PASS NMR.
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Chapter 2
N M R Spectroscopy
NM R spectroscopy involves the m anipulation and detection of
nuclear m agnetization. The
nuclear m agnetic moment (/x) of a nucleus is proportional and
collinear to the quantized
spin angular m om entum (Ift),
H = 7 M , (2.1)
where h — 1.0545727 x 1 0 34 J s is P lanck’s constant and 7 is
the magnetogyric ratio. Only
those nuclei w ith non-zero spin quantum num ber have m
agnetization and can be detected.
If A (mass num ber) and Z (charge number) of a nucleus are both
even, then the nucleus has
7 = 0; for example, 12C, 160 , and 32S. If A is even and Z odd,
the the nucleus has integral
spin; for example, 2H and 14N have 7 = 1. If A is odd, then the
nucleus has half-integral
spin; e.g. 4H and 13C have 7 = 1/2, and 170 has I = 5/2. The
spin quantum number,
magnetogyric ratio, and natu ra l abundance of several nuclei
are listed in Table 2.1.
In an external m agnetic field Bo, the potential energy of
orientation of a nuclear moment
splits into 2 1 + 1 Zeeman levels,
E m = - n - B 0 = - ^ h m B o , (2.2)
where the m agnetic quantum num ber m takes values —7, —7 + 1 ,
. . . , / — 1,7. The energy
4
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CH APTER 2. NM R SPEC TRO SC O PY
T able 2.1: M agnetic properties of a few nuclei [109].
Nucleus I 7 (107 rad-T 4s 7) Freq. (MHz)a Abund. (%)
Receptivity^!H 1/2 26.7522128 300 99.985 1.02h 1 4.10662791 46.05
0.015 0.0000014513c 1/2 6.728284 75.44 1.10 0.000175170 5/2
-3.62808 40.67 0.038 0.000011123Na 3/2 7.0808493 79.36 100
0.092725Mg 5/2 -1.63887 18.36 10.0 0.00026845Sc 7/2 6.5087973 72.88
100 0.30293Nb 9/2 6.5674 73.43 100 0.488207P b 1/2 5.58046 62.76
22.1 0.00201
“ The resonance frequency at 7.0459800 Tesla.6 The receptivity
is proportional to the cube of 7 tim es the natural abundance.
difference between two adjacent levels is A E — Ii.uq, where
j B q27r
(2.3)
is the Larm or frequency. W hen excited by radio-frequency (rf)
irradiation, the atom s will
resonate, or absorb energy at this frequency. The detected
resonance frequency identifies the
kind of atom and the other atom s to which it is connected in
the molecule. The molecular
structure can be determ ined by m easuring all of the
frequencies.
The system is a t finite tem perature; thus spin populations of
these levels are not equal
bu t are instead distribu ted according to the Boltzm ann law of
statistical mechanics N m =
N exp (—Em/ k T ), where N is the to ta l num ber of spins. The
net macroscopic m agnetization
is
Y lm -1 r m expPyhmBo/ kT) M = "j:- — ------- --------' n r a =
/ (2.4)E ™ = -/ ex p ( ih m B 0/kT)
For proton 'yh/k = 0.002K/Tesla, therefore the ratio j h B o / k
T is always small for currently
achievable static m agnetic fields (< 45 Tesla) a t tem
peratures above IK . This suffices to per
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CHAPTER 2. N M R SPEC TRO SC O PY 6
m it a linear expansion of the Boltzm ann exponential ( “high
tem peratu re” approxim ation)
to obtain
3kT K 1
At room tem perature in an 7 Tesla m agnetic field, there are
only about 50 more protons
in the spin-up sta te th an those in the spin-down sta te out of
a to ta l 106 protons. It is this
tiny population difference th a t is responsible for the entire
NM R signal.
This chapter only covers a few aspects of NM R spectroscopy. For
more inform ation
these textbooks are good sources [1, 45, 95, 39, 75, 79, 81,
89].
2.1 P recessin g M a g n etiza tio n and N M R Signal
In a static field Bo, the nuclear m agnetic moment precesses a t
an angular velocity
uj0 = —yB 0 , (2.6)
its m agnitude is 27r tim es the Larm or frequency. For positive
7 , the negative loq means the
m agnetic moment precesses clockwise around —Bo- The macroscopic
m agnetization also
undergoes the same precession. To effectively pertu rb the
system, a small m agnetic field
2j3i c o s {urf t + 4>o) oscillating at a frequency very
close to the Larm or frequency (u,yy wo)
is applied perpendicular to Bo (Fig. 2.1). Since negative
frequency doesn’t have special
meaning in electronics, the signed u)rf here should be
interpreted as its absolute value when
used in the sense of electronics. This field consists of two
oppositely ro tating circular mag
netic fields, B\exp{iu>rft ) and B \ exp(—iu>rft) (drop
4> 0 here and below). The component
B\ exp (—icorf t) has an angular velocity of about 2u>o
relative w ith the precessing spin, this
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CH APTER 2. NM R SPEC TRO SC O PY 7
component barely affects th e nuclear spin. The other component
B \ exp(itorf t ) ro ta tes at
an angular velocity very close to the Larmor precession and can
thus causes stim ulated
absorption of energy. It is convenient to change to a frame ro
tating at speed u rf w ith the
laboratory frame. In this ro ta ting frame, the B\ field appear
static and it exerts a torque
to bring the m agnetic m oment toward the x'y' plane a t an
angular velocity
LOi = j B i , (2.7)
as depicted Fig. 2.1. Usually the rf irradiation is tu rned off
after tim e T„y2> which satisfies
ujit 7[/ 2 — vr/2, the m agnetic moment is tipped onto the x'y'
plane; th is rf pulse is referred to
as 7t/2 or 90° pulse. Viewed from the laboratory frame, the
magnetic moment precesses in
the x'y1 plane at the Larm or frequency. This causes the
magnetic flux in the coil to change
w ith tim e and induce a voltage. This voltage is detected and
amplified to provide the NM R
signal.
electriccurrent
X '
F igu re 2.1: C reation of B \ field in the laboratory frame
(left) and the effect of rf pulse in the ro tating frame (right).
In the ro tating frame, the axis labels are primed.
The acquired signal in tim e domain, which is usually called the
free induction decay
(FID), is shown Fig. 2.2. T he corresponding frequency dom ain
spectrum is obtained by
Fourier transform ation. T he FID may be expressed as
Sit) = 5oe
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CHAPTER 2. NM R SPECTRO SCO PY 8
where t > 0 , the relative precession frequency in the ro
tating frame is defined as Q =
cjq — ujrf. and T | is the transverse relaxation tim e which
shall be described shortly in
section 2.4. By carrying out a Fourier transform f S(t)
exp(—iu>t)dt the frequency dom ain
spectrum is
I ( oj) = S'o[a(ca) + id(uo)\, (2.9)
where
“ ^ = (fi - a;)2 + R 2 ’ d^ = (Q - oj) 2 + R 2 ’ (2 ‘10)
and R = 1 /T | is the transverse relaxation rate. Both the real
and im aginary com ponents of
the spectrum are plotted in Fig. 2.2. The real, absorptive
component is narrow and it has
a full-width of 2 /T | a t the half-maximum. The imaginary,
dispersive component has much
wider linewidth; serious overlap occurs if there is more th an
one resonance peak. Therefore,
the absorption lineshape provides be tte r resolution and is
usually presented.
-t/T*
time, t frequency, o)/2 ti
F igure 2.2: Free induction decay w ith envelope (left) and
spectra (right). The real spectrum is narrow and absorptive (solid)
and the imaginary spectrum is wide and dispersive (dashed). The
full-width a t half-height of the absorption line is 2/T f .
2.2 D en sity M atrix
In an NM R experim ent, to theoretically describe the tim e
evolution of an ensemble of
nuclear spins, the quantum mechanic density m atrix formalism is
necessary. In a mixed
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CH APTER 2. NM R SPEC TRO SC O PY 9
ensemble, the expectation value of an arb itrary operator O
is
= (2 .H )i
where u>i is the fractional population of the sta te Define
the density m atrix as
p = (2.12)i
w ith the m atrix elements given by
(^k\p\tpj) = . (2.13)i
Using the closure relation the expectation value in Eq. (2.11)
becomes
(O) = Y^^i(i’i\^j){^j\o\i>k)(i>k\i’i)i j k
i j k
= ^2(i’k\p\^j)(^j\o\^k)j k
= T r (p O ) , (2.14)
where T r means evaluating the trace.
In the Zeeman representation, the diagonal elements pa of the
density m atrix are pop
ulations of the eigenstates. Off diagonal elements pij (i / j )
describe the phase coherence
of the spins and
p = mi — rrij (2-15)
is called the order of the coherence, where mi and rn:l are the
m agnetic quantum num bers
of the |ipi) and \yj.j) states, respectively. Only the ± l-q u
an tu m coherence can be directly
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CH APTER 2. NM R SPECTRO SCO PY 10
detected due to the selection rules; conventionally the
—1-quantum coherence is detected in
NMR. However, m ulti-quantum coherence may be indirectly
detected in multi-dim ensional
NM R experim ents (such as 2D-PASS and MQMAS experiments in C
hapters 3 and 4 respec
tively). In an multi-pulse NM R experiment, a specific coherence
transfer pathw ay needs to
be selected; the guidelines for the pathway selection are
described in A ppendix B.
The tim e evolution of the density m atrix is determ ined by the
Liouville-von Neum ann
equation,
The classic ro ta ting frame is equivalent to the interaction
representation in quantum me
chanics, where the density m atrix and H\ become
For simplicity, we drop the tilde and imply working in the
interaction representation; also the
(2.16)
and the H am iltonian H usually consists of the large and tim e
independent Zeeman interac
tion H z and other interactions denoted by H \ ;
H = Hz + H i . (2.17)
H x = (2.18)
and the equation of evolution in the interaction representation
is
(2.19)
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CHAPTER 2. NM R SPECTRO SCO PY 11
subscript of Hi is dropped. The Liouville-von Neum ann equation
may be solved iteratively,
1 dt! f [[p{Q),H{t'%H(t'))dt" , (2.20)Jo
where orders higher th an the second are truncated . However,
for numerical sim ulation it is
be tter to use the propagator formalism,
p(t) = U ( t ,0 )p (0 )U \ t ,0 ) , (2.21)
where U(t, 0) is the un itary propagator responsible for the
spin dynamics in the period from
0 to t. The propagator is defined as
U(t,0) = T e x p j - i ^ H(t ' )d t '^ (2.22)
w ith T being the Dyson tim e-ordering operator for Ham
iltonians containing non-com m uting
components. In practice th is is usually evaluated by a simple
tim e-ordered produt
n —1
U(t, 0) = U e x p { - i H ( j A t ) A t ] , (2.23)j=o
where n is the num ber of infinitesimal tim e intervals A t
during which the H am iltonian may
be considered tim e independent. The exponentials are evaluated
by diagonalization of the
m atrix representation of the Ham iltonian. A versatile sim
ulation program for solid-state
NM R spectroscopy, S IM P S O N , is becoming popular [8].
2.3 N M R S p ectro m eter
An NM R spectrom eter m ust be capable of m anipulating rf pulse
and detecting nuclear
m agnetization. M ajor components of a m odern pulsed NM R
spectrom eter are shown in
p(t) = p(0) + i [ [p(0),H(t')]dt' - j Ji Jo
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CHAPTER 2. NM R SPECTRO SCO PY 12
Figure 2.3: superconducting magnet, probe, frequency
synthesizer, pulse program m er, R F
transm itter, receiver, digitizer, and computer.
Pulse Programmer RF Transmitter
Frequency Synthesizer
Digitizer Receiver
Maunet
rlx
F igu re 2.3: An NM R spectrom eter.
The NM R m agnet provides a strong, stable (0.01 ppm /hour for a
7 Tesla OXFORD
m agnet), and extrem ely homogenous (0.1 ppm half-height line w
idth w ith superconducting
shims for given sample volume) m agnetic field into which the
sample is placed. The strength
of the NM R spectrom eter is typically specified in term s of
the resonance frequency for proton
expressed in MHz. The highest static m agnetic field achieved to
date is 45 T in the National
High M agnetic Field Lab (NHMFL), Tallahassee, FL. Once in
operation, this m agnet, which
is a hybrid of electro- and superconducting types, uses nearly
70% of the electric power in
the city of Tallahassee. Superconducting m agnets are desirable
since after being energized
they consume no electric power. Both Oxford Instrum ents and
Bruker BioSpin have recently
commercialized their highest field superconducting 900 MHz (21.1
T) magnets.
T h e NM R p ro b e ho ld s th e sam p le a n d is p laced in th
e b o re of th e m a g n e t. T h e p ro b e
also contains one or two coils for irradiating the sample w ith
rf energy and for detecting the
very weak nuclear signal from the sample. There are special
probes, such as those capable
of variable tem perature, of ro ta ting the sample around an
axis m aking a fixed or variable
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CH APTER 2. NM R SPECTRO SCO PY 13
angle w ith the sta tic field. One of the latest inventions for
liquid s ta te NM R spectroscopy is
the cryogenic probe. W ith the electronics working a t low tem
perature, the reduced therm al
noise allows high sensitivity to be achieved.
The NM R console consists of pulse program m er, rf transm
itter, frequency synthesizer,
receiver, and digitizer. It generates and controls rf pulses
used to excite the sample in the
probe. The console also receives and amplifies the very weak
signals coming back from the
probe.
NMR spectrom eters also rely on com puters and software for the
control of the various
pulse sequences and the collection, processing, and storage of
the NM R data. The NM R
signals are subjected to complex digital signal processing
algorithm s including the Fourier
Transform to convert the tim e dom ain FIDs into frequency dom
ain spectra for in terpreta
tion.
2.4 R ela x a tio n T im es
In a static m agnetic field, the m agnetization is quantized and
oriented along the ^-direction
(parallel to the field). A n/2 pulse results in zero
longitudinal m agnetization, M z = 0.
After the rf pertu rbation is removed, the nuclear spins release
energy to the surrounding
environm ent and M z recovers according to,
and T\ is the tim e constant. In reality, non-exponential
recovery may also be observed.
Given long enough tim e, therm al equilibrium will be
re-established and the longitudinal
m agnetization returns to its equilibrium value, M z {oo) = M
q.
M z (t) = M 0( l - e - * / Tl) (2.24)
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CHAPTER 2. NM R SPECTRO SCO PY 14
Similarly, the decay of transverse m agnetization is called
transverse relaxation. After a
7r/2 pulse the transverse m agnetization decays toward its therm
al equilibrium value of zero.
Inevitably, the transverse relaxation tim e is shorter th an the
longitudinal relaxation time.
In addition, factors like field inhomogeneity, dipolar coupling,
chemical shift anisotropy,
and quadrupole interaction cause components of the transverse m
agnetization to fan out,
producing additional dephasing mechanisms for the net transverse
m agnetization. The
contribution from field inhomogeneity is characterized by the
tim e constant T'2 (~ I / 7 AZ?).
Symbol I 2 is used to denote the intrinsic tim e constant
arising from all o ther factors. The
overall effective relaxation tim e T | is calculated according
to
J _ - i _ 1T* ~ T~2 + T /̂ * (2.25)
In nonviscous liquids, fast m otion averages out all the
anisotropic interactions, resulting in
1 2 = T ]. However in solids (and some liquids), T2 is much
shorter th an T).
saturation comb
.V £ ✓— * t—1k " " ‘F igure 2.4: T\ m easurem ent using sa
turation comb. The satu ration comb consists of 4-20 7r/2 pulses
separated by interval r , T2* < r
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CH APTER 2. NM R SPECTRO SCO PY 15
saturation, th e interval r between two pulses should satisfy T£
< r < Tj. A nother tt/2
pulse is applied after a delay f, and the signal is then
acquired. The signal M (/,) obeys
M{t) = Mq(1 - e~t/Tl) . (2.26)
M(
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CH APTER 2. NM R SPEC TRO SC O PY 16
which is established by an rf pulse in conductive probe
components in the m agnetic field,
is converted back to an rf radiation by the reciprocal mechanism
and interferes w ith the
NMR signal (acoustic ringing down problem).
F igure 2.5: Spin echo pulse sequence and vector diagram s to m
onitor the m agnetization. ‘S’ stands for m agnetization species w
ith slow Larm or precession, ‘F ’ stands for fast precession.
The deadtim e problem is usually solved by using spin echo
techniques. In the pulse
interval r ; the subscripts denote the phases of the applied
pulses: x for 00 = 0 and y
for o = 7t / 2. The iry pulse refocuses the decayed FID after
tim e r ; the mechanism of
refocusing is illustrated in the vector diagrams. For
simplicity, suppose the m agnetization
consists of only two components, one w ith slow precession
speed, another w ith fast speed.
The (tt/2 )x pulse brings the m agnetization onto the y axis.
The components then fan out
(or dephase); the fast component move clockwise while the slow
component moves counter
clockwise. Their vector sum, the transverse m agnetization,
decays. The Try pulse flips
the two com ponents by 180° around the —y axis, and then each
com ponent continues its
journey in its original direction. As a result, the two
components come closer and a t tim e
r after the second pulse, they meet to give a m aximum m
agnetization. Then they part and
the m agnetization decays. By choosing r longer th an the
deadtim e, the above m entioned
(71/2)*
T
sequence shown in Fig. 2.5, a ixy pulse is applied after the
initial (n / 2) x pulse w ith tim e
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CH APTER 2. NM R SPEC TRO SC O PY 17
problems are avoided. Spin echoes have become im portan t
components in many sequences
for solid sate NM R spectroscopy.
2.6 Sum m ary
This chapter introduces NM R spectroscopy from bo th a classic
description of the precessing
m agnetization and a quantum mechanic density m atrix
description. The construction of a
m odern NM R spectrom eter is also briefly covered. The m ethods
of relaxation tim e mea
surement are described. The satu ration comb m ethod will be
used to m easure the 207Pb
longitudinal relaxation tim e for relaxor ferroelectric m
aterials in C hapter 8 . The im por
tan t spin echo concept is also introduced. This solves the dead
tim e problem and can be
combined w ith MQMAS and 2D-PASS sequences th a t appear in
later chapters.
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Chapter 3
Chemical Shift Interaction
Consider a nucleus subject to a static external m agnetic field
Bo- The local m agnetic field
(B) it in teracts w ith differs from Bo, since the external
field is shielded by the surrounding
electrons. This “chemical shielding” has bo th diam agnetic and
param agnetic contributions.
The diam agnetic shielding is described in term s of Lenz’s law;
the electrons circulating
around the nucleus induce a small field opposite to the applied
field. The param agnetic
shielding is associated w ith excited electronic sta tes and
results in a small field along the
applied field. Formal theory of the chemical shielding
interaction can be found in Slichter’s
book [95].
The chemical shielding interaction is influenced by surrounding
atom s and chemical
bonds. Therefore, the nuclei can be used to probe the chemical
environm ent. One classic
example is the proton NM R spectrum of ethyl alcohol
(CH3-CH2-OH), which is composed
of three set of peaks w ith different chemical shifts. These are
assigned to methyl, methylene,
and hydrooxyl protons and have integrated intensities
proportional to the num ber of protons
in each site, i.e., 3:2:1.
18
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CH APTER 3. CHEMICAL SHIFT IN TE R A C TIO N 19
3.1 C hem ica l S h ield in g and C hem ical Shifts
The local field experienced a t the nucleus is given by
B = ( 1 - < t)B 0 , (3.1)
where a is the shielding tensor. It is possible to choose a
principal axes system (PAS) where
only the diagonal components are nonzero. These diagonal
components
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N 20
to ensure ft > 0 and —1 < k < +1 (k = —1 when $22 = £33
and n = +1 when 8 2 2 = $11)-
However, it is im portant to be aware of another convention and
to be able to convert
between the two. This earlier convention uses the anisotropic
chemical shift $ a niso instead
of the span f t , and the asym m etry param eter 77 instead of
the skew k ,
$ an iso — $33 $iso >
$11 - $22 , . V = J , (3-4)
^33 ^ iso
where the three com ponents should be so sorted |$33 — $ is o |
> |$22 — $ iso | > |$n — $ iso | as to
ensure 0 < r/ < 1. This ordering places $n between 8 2 2
and $ 3 3 , and closer to $22- In the
case of axial sym m etry (77 = 0 ) , notations $ || = $33 and $
x = $11 = $22 are also used widely
in the literature. The formulas to convert them to the first
convention are,
0 ir i 3 + 771 I | Oj“'amso2 ’
3n — 3k = Sign(8 an-lso) —-- — . (3.5)
•0 ~v Tj
And the inverse relations are,
$aniso = ~Sign(n) ft 3 + ^6 ’
3 _ 3 IKI ,on \71 = J T M ■ (3-6)
The second convention will be used in this thesis. But adding to
the chaos of two
conventions, we m ay find in the literature a slightly different
definition of 77 for the second
convention [cf Eq. (3.4)],
_ $22 - $11 -s.71 ~ $33 - $iso ’ ( }
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CHAPTER 3. CHEMICAL SH IFT IN TERA C TIO N 21
w ith ordering of com ponents | (̂ 33— ̂ iSo | > |$li — $ so|
> j^22 —
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CHAPTER 3. CHEMICAL SHIFT IN TERA C TIO N
In the PAS frame, these reduce to
22
ro = Sa
p pf ± i = 0 , (3.12)
f ± 2 = T/daniso/VQ .
The components in the PAS frame (P) are conveniently transform
ed into the labora
tory frame (L) by W igner ro tation m atrix T>2 (FLpl)
through Euler angles 14 p l = (ck,/3,7 )
(Appendix A),2
f m = E ^ 2m' m ^ P L ) C . (3-13)m ' = - 2
Im m ediately we have
4 = fo = S,L _
'an iso3 cos2 (3 — l + rj sin2 /? cos 2a
(3.14)
and
, 3 cos2 (3 — 1 + r] sin2 (3 cos 2aU7.7. = " is o 4~ " a
(3.15)
In a powder sample, the crystallites are random ly oriented. The
orientation dependence
of the chemical shift [Eq. (3.15)] results in anisotropic
lineshape w ith two edges and a
peak a t frequencies equal the three principal values
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N 23
Static MAS
Experiment
*. . . i i
Simultaion
-3400 -3600 -3800 -4000 -3400 -3600 -3800 -4000Frequency (ppm)
Frequency (ppm)
F igu re 3.1: Experim ental and sim ulated s ta tic and MAS 207P
b NM R spectra of PbS04 at 7 T. The scaling for each sub-figure is
exactly the same. The static spectrum was acquired w ith recycle
delay 30 s, dwell tim e 20 /rs, and 1312 scans. The MAS spectrum
was acquired w ith a sample spinning ra te of 5.0 kHz, recycle
delay 30 s, dwell tim e 10 (is, and 1168 scans. Spectra were sim
ulated w ith ^ so = —3607 ppm,
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N 24
many spinning sidebands are observed (separated by lur/2 7r). If
the sample spinning is
slow, these narrow lines have an envelope close to the static
lineshape. An example of MAS
powder p a tte rn split into spinning sidebands is shown in Fig.
3.1 for PbSC>4.
The intensities of the sidebands can be calculated rigorously
using the following MAS
theory. The irreducible spherical tensor can be w ritten as
2 2f m = £ V l 'm ^ R L ) £ V 2m„ml{0PR) C , , (3.16)
m '=—2 m "=—2
where Euler angles f lpR = (a, [3,7 ) transla te the PAS frame
to rotor frame (R), and Euler
angles FLrl = (oJRt, a rc c o s (l/v /3), 0) transla te the
rotor frame to LAB frame. For simplicity
we drop the tim e-independent isotropic part, so the observed
frequency (in Hz, cul/IO6 factor
since /o ' is in ppm) is
= Ci cos(wr£ + 7 ) + S i sin (coRt + 7 )
+C 2 cos(2ujRt + 27) + S 2 sin(2u>Rt + 27) , (3-17)
where the four coefficients Ci, Si, i = 1,2 are functions of the
orientation of the PAS and
rotor frames {a,(3). In some situations, it is more convenient
to use an equivalent form,
u(t) = C\ cos(ujRt) + S 1 sin(u7jf) + C 2 cos(2ujRt) + S 2
sin(2n>jR£), (3.18)
where the four coefficients Ci, Si, i = 1,2 are functions of (a,
(3,7 ) [91].
For a single crystallite, the MAS tim e dom ain signal is
s(t) = e^o “'(*')*' = ei m e- i m ) (3 .19)
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CH APTER 3. CHEMICAL SH IFT IN TE R A C TIO N 25
w ith phase
$ (« ,/? , 7 + u;Rt) = J w (a ,/3 ,7 + u R t ) d t= — [2C±
sin{lorJ) — 2 S i cos(a)R t) +
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CH APTER 3. CHEMICAL SH IFT IN TERA C TIO N 26
This sideband intensity is for a single crystal or crystallite,
and is a complex valued function.
For powder samples, the next step is to integrate over 7 ,
IN (a, (3) = IN (a, P , l ) ~ = G fr(a , f3)GN (a, 0 ) .
(3.25)
It is real and positive. The observed iVth sideband intensity of
a powder sample,
I n = da In (&, P) sin/3d/3, (3.26)Jo Jo
is a function of Ci, Si, (* = 1,2); the integrations over a and
(3 preserve the real and positive
property (consult Schm idt-Rohr’s book for a rigorous proof
[91]).
From the intensities of these sidebands, the principal values of
the chemical shift tensor
can be calculated. Hertzfeld and Berger [56] plot Tn /T q
contour m aps for the first few
sidebands w ith regard to two variables p and //,
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N 27
are often acquired to differentiate isotropic peaks from
sidebands. However, confusion is
unavoidable if there are too many inequivalent sites. Moreover,
the distribution of intensity
among the sidebands results in low sensitivity for large
CSA.
From these considerations, it seems advantageous to suppress the
sidebands and effec
tively achieve the infinite spinning speed limit spectra. In
1982, Dixon proposed the Total
Suppression of Sidebands (TOSS) sequence, which consists of four
appropriately spaced n
pulses, to achieve to ta l suppression of spinning sidebands
[37], The sidebands are suppressed
by inhibiting ro tational echoes from forming. The TOSS signal
can be expressed as
s(t) = g{7 + u Rt ) . (3.28)
At no point in tim e do the m agnetizations from all the
different crystallites coherently
refocus, thus no ro tational echoes will form [cf Eq. (3.22)].
The sideband intensity of a
single crystallite is [cf Eq. (3.23)]
I N (a ,P ,1 ) = ei^NGN (a,(3). (3.29)
For a powder sample, the 7-averaged intensity is
I N (a, (3) = GN (a, (3) [*" = GN (a, /3)SNfl , (3.30)Jo 7̂r
where Sn ,0 is Kronecker delta. Thus only the center band is
preserved and all the sidebands
are removed.
The centerband intensity for ordinary MAS is the (a,/3)-averaged
G*N(a, f3)G/y (a , (3),
which is always real and positive. However, the TOSS intensity,
which is the averaged
G/v(«) (3), could be complex. This is one disadvantage of the
TOSS m ethod.
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CHAPTER 3. CHEMICAL SH IFT IN TERA C TIO N 28
3.5 P h a se A d ju sted S p in n in g S idebands
TOSS achieves high resolution at the expense of im portant
chemical shift anisotropy infor
m ation. Is it possible to preserve the anisotropic inform ation
and a t the same tim e achieve
high resolution? In 1995 an excellent solution was introduced—
2D-PASS (two-Dimensional
Phase-A djusted Spinning Sidebands), by Antzukin, Shekar and
Levitt [7] on the basis of
Dixon’s other rem arkable m ethod, PASS [36, 37]. The 2D-PASS
separates sidebands by
their order along the uq dimension, while the uq dimension
consists of bo th isotropic and
anisotropic shifts. The original spectrum of L-tyrosine
hydrochloride is reproduced from
A ntzutkin et. al. in FIG . 3.2. A simple shearing transform
ation removes the anisotropy
in the oq dimension so th a t anisotropic and isotropic shifts
are cleanly separated along
two orthogonal axes. (An example of shearing transform ation is
shown a t the end of this
section.)
The pulse sequence for 2D-PASS is shown in FIG. 3.3. Following
the initial tt/2 pulse,
five 7T pulses are applied a t tim es Q (* = 1 • • • 5). The
period 0 T, which is called PASS
element, is divided into six intervals r , (i = 1 . . . 6).
Usually, T = tn, bu t integer num ber
of rotor periods may be added to 75 and tq in order to create
spin echoes [41]. D ata
acquisition sta rts immediately after the PASS element.
Application of a n pulse inverts
the sign of precession frequency. Therefore, the effective phase
for the isotropic part, which
should rem ain unchanged at the end of PASS element (to ensure
th a t uq dimension contains
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N 29
HClC nh„ci
m l 0 CDi/ oOr
-6
- s-10 -5 0 5
© 2/271 (k H z)10
F igure 3.2: Exam ple spectrum of 2D-PASS from A ntzutkin et al.
[7]. Top: 13C MAS spectrum of L-tyrosine hydrochloride powder a t
sample spinning speed of 1030 Hz, 3888 transients were collected.
Bottom : 2D-PASS spectrum showing the separation of sidebands in
the uq dimension. T he uj 1 slices are labeled w ith the order of
sideband. Sixteen t\ increments were taken, each w ith 243
transients. (Notice the frequencies are labeled in the opposite
direction to NM R convention.)
only the anisotropy), is
0 = W iso[(T 6 + T4 + r2) - (7 5 + 7-3 + Ti)]
,[T + 2 £ ( - ! ) % * ] .— C
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CHAPTER 3. CHEMICAL SH IFT IN TERA C TIO N 30
tl4 -T
t i
1 Xl ■ X2 ■ X3 j X4 | 15X5 ■ X6
F igure 3.3: Pulse sequence and coherence pathway for 2D-PASS
experiments. The initial 7t/2 pulse is followed by five
appropriately spaced 7r pulses.
For the anisotropic part, the effective phase a t t 2 is
r Ci r C2 K3 Ka Ks rT+t2= — / uj(t)dt + / uj(t)dt — / Lo{t)dt +
/ u>(t)dt — / ui{t)dt + / u>(t)dt
JO Jo JC2 J Cs J(a JCs
fT+t 2
R l JC.2 JC.3 JC.A J C s
= $ (T + t 2) -2 $ (C 5 ) + 2 $ (C 4 )-2 $ (C 3) + 2‘l>(C
2)-2$(C i) + $ (0 ) . (3.32)
The same phase can be created by a free evolution (without rf
pulses) starting from an
effective tim e t\ before the end of th e period T,
* = $ ( r + *2) - $ ( r - t 1). (3.33)
We define 0 = uir{T — t\), which is dubbed the “pitch” by Dixon
[36]. The above two
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CH APTER 3. CHEMICAL SHIFT IN TERA C TIO N
equations, w ith definition of 4>(t) in Eq. (3.20), yield
31
- S i
+ a2
S 2 ' 2
C i 2 ^ ] ( - l ) n s i n ( u j R Cn) + sin (0 )_ n=l
5
2 ^ ( - l ) n c o s ^ C n ) + cos(0 ) + 1 ^=l ' 5
2 ^ ( - 1 ) ” sin(2wfiCn) + Sin(20). n=1
2 y ^ ( - i r cos(2 lorCu) + cos(2 0 ) + 1 = 0 .
(3.34)
This equation is valid for any orientation, therefore all four
expressions in square brackets
m ust vanish. These four equations plus Eq. (3.31) are solved to
obtain the five Q (and
hence t*) values for a given 0 . Listed in Table 3.1 are
solutions for © increment of t R j 16.
T able 3.1: Delays between pulses for 2D-PASS sequence shown in
FIG . 3.3.Increment 0 / 2 tr n / t R T 2 / t R r z / t R T 4 , / t
R T 5 / t R T e / t R1 0.00000 0.16667 0.16667 0.16667 0.16667
0.16667 0.166672 0.06250 0.18989 0.17512 0.15541 0.17615 0.15470
0.148733 0.12500 0.21668 0.17874 0.14270 0.18365 0.14062 0.137614
0.18750 0.24453 0.17549 0.12896 0.18947 0.12651 0.135045 0.25000
0.26915 0.16096 0.11573 0.19608 0.11511 0.142976 0.31250 0.28147
0.12931 0.11053 0.21004 0.10800 0.160657 0.37500 0.26731 0.09195
0.13166 0.22798 0.10102 0.180088 0.43750 0.23472 0.07687 0.17459
0.22846 0.09069 0.194679 0.50000 0.20979 0.08043 0.20978 0.20979
0.08043 0.2097810 0.56250 0.19467 0.09069 0.22846 0.17459 0.07687
0.2347211 0.62500 0.18008 0.10102 0.22798 0.13166 0.09195 0.2673112
0.68750 0.16065 0.10800 0.21005 0.11052 0.12931 0.2814713 0.75000
0.14297 0.11511 0.19608 0.11574 0.16095 0.2691514 0.81250 0.13504
0.12651 0.18947 0.12896 0.17549 0.2445315 0.87500 0.13761 0.14062
0.18365 0.14270 0.17875 0.2166716 0.93750 0.14874 0.15469 0.17615
0.15541 0.17512 0.18989
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CH APTER 3. CHEMICAL SHIFT IN TE R A C TIO N
The signal can be w ritten as
32
s ( t u t 2 ) =
= g \ i + u R{ T - t l ))g { 1 + ujR{T + t 2 ))ei^ e - x{T+t^ .
(3.35)
The signal is periodic w ith the rotor period t R in bo th
dimensions, and thus intensities
appear only a t multiples of lor . For simplicity, ignoring the
relaxation and isotropic factors,
double Fourier transform ation of the signal gives the
intensity
[ s ( h , t 2 ) e - i N l j R t 2 — . (3.36)tR J tR
W ith substitu tions 7 ' = 7 + ujr{T — t \ ) and 7" = 7 + lor (T
+ t 2 ) ,
Si 7) = ei(N~M)0+wRT) j / d̂ ~ iNi'g{i")= ei(-N^ M^ +^ G * M(a
,S )G N (a,(3). (3.37)
Averaging over 7 gives
(3) = J A 7 ) ^ = &m ,n I n { (3.38)w ith Fv defined in Eq.
(3.23). The Kronecker S m ,n means th a t sideband of order N
only appear in the iVth u \ slice. And the intensity of the
sideband is the same as in the
conventional MAS spectrum .
To reduce pulse imperfections and to select the coherence pathw
ay 0 —> + 1 —> —1 —>
+ 1 —> —1 —> + ! —> —1, A ntzukin et al. [7]
independently phase cycle the 7r pulses in three
I m ,n = j
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CHAPTER 3. CHEMICAL SHIFT IN TERA C TIO N 33
step of 27r/3. This results in long phase cycling of 243 to ta l
steps even w ithout considering
the four-step (of 7r / 2) cycling of the initial tt/2 pulse. In
fact, it can be reduced to only
three steps, as in Table 3.2. The simplification has been made
by cycling the first three and
the last two n pulses in a coordinated way,
t l ~ (f>2 + t s = 0° (or 360°),
-< k + t s = 0° , (3.39)
where t i (?' = 1 . 2 , . . . , 5). The same coherence transfer
pathway is selected by bo th the
original and this simplified phase-cycling schemes. The new
phase cycle saves experiment
tim e (see Appendix B). Four-step cycling of the 7r/2 pulse can
also be used to further
remove imperfections. Even w ith four-step cycling, there are
only twelve steps in total!
T able 3.2: Improved phase cycling for 7r pulses in 2D-PASS
(unit in 27t/3).
t l t% t3 tA tb0 0 0 0 01 2 1 1 12 1 2 2 2
Direct Fourier transform ations along bo th dimensions result in
a spectrum where side
bands (and centerband) belonging to one site are distributed
along a diagonal line (using
the same unit for bo th dimensions), see Fig. 3.4(a). In fact,
the loi dimension contains only
anisotropic shifts, while the u? dimension contains bo th
isotropic and anisotropic shifts. A
shearing transform ation is usually applied so th a t the ui-2
dimension contains only isotropic
shifts. Now sidebands belonging to one site are parallel to uj\
axis and the isotropy and
anisotropy are fully separated [see Fig. 3.4(b)]. The separation
of the isotropic shift gives
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CH APTER 3. CHEMICAL SH IFT IN TE R A C TIO N 34
high resolution, while the separation of anisotropy preserves
valuable inform ation of chem
ical shielding tensor elements. A ntzukin et al. [7] perform the
shearing transform ation by
m ultiple left shifts. Left shift by an integer num ber of
points, however, introduces errors
due to insufficient digital resolution. The problem can be
solved using “fractional left shift”
and this is accomplished by linear phase correction in the tim e
domain. F irst, inverse F T
in the u>2 dimension; for each slice, m ultiply with sideband
order M ; then the
second F T w ith respect to t\ is applied.
--300 -300
-300 -300
-4000-4000-3300 -3300 Isotropic (ppm)MAS (ppm)
F ig u re 3.4: 2D-PASS 207P b NM R spectra of PbSCL at 7 T. The
sample spinning speed was 5.0 kHz, recycle delay was 30 s, dwell
tim e was 20 ps, and 180 scans for each pitch increment. No
shearing transform ation was applied to spectrum in (a); spectrum
in (b) was sheared. Slice taken at
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CH APTER 3. CHEMICAL SHIFT IN TE R A C TIO N 35
mg tensor is characterized by isotropic and anisotropic chemical
shifts. For chemical shift
anisotropy, the two conventions—fI, n versus (5anjso, rj— are
described and conversion for
mulas are provided. For powder samples, b o th static and MAS
lineshapes are calculated
starting from the chemical shift Ham iltonian; this allows fit
to experim ental spectra and
extraction of the principal elements of the chemical shielding
tensor. The 2D-PASS tech
nique separates isotropic chemical shift into the x'-dimension
and the anisotropic pa tte rn
of spinning sidebands into the y-dimension; th is achieves high
resolution and at the same
tim e obtains all principal elements of the shielding tensor.
For disordered m aterials such as
the relaxor ferroelectric (1 — x) PbM g1/ 3Nb2/303 - x P b S c ^
N b ^ O s we are going to study
in Chapter 8 , bo th sta tic and MAS 207Pb spectra have such low
resolution th a t they could
not provide much insight into local structures. Only the
high-resolution 207P b 2D-PASS
experiments prove to be indispensable for the study of such
disordered m aterials.
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Chapter 4
Electric Quadrupole Interaction
4.1 Q uadrupolar N u cle i and Q uadrupolar In teraction
As we noted in chapter 2, most elements have nuclear m agnetic m
oments as a result of non
zero nuclear spin and thus in principle are detectable through
NM R spectroscopy. C hapter 3
concentrated on the chemical shift effects arising from the
interactions between the m ag
netic moments and the local m agnetic fields. In addition, most
of these m agnetically active
elements possess electric quadrupole moments arising from a
nonspherically sym m etric nu
clear electronic charge distribution. Reorientation of a
spherical nucleus in a surrounding
electric field does not change the electrostatic energy. Nuclei
w ith spin 1 = 0, having no
preferred nuclear orientation a t all, possess no electric
quadrupolar moment. Moreover,
flipping spins of I = 1/2 does not change the charge
distribution, and therefore, such nuclei
do not display the quadrupolar effect either. Only I > 1/2
nuclei can have quadrupole
m o m en ts . B y c lassify ing th e 103 e lem en ts acco rd in
g to th e sp in o f th e ir iso to p es , we find
th a t there are only twelve having 1 = 0 isotopes, 25 elements
having 7 = 1 / 2 isotopes,
and 71 having I > 1 isotopes. Of these 71 elements 69 have
isotopes w ith half integer spin
n + 1/2, (n = 1, 2, . . . ) . T ha t is to say, two-thirds of
the elements have nuclear quadrupole
36
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CHAPTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 37
moments. These elements are ubiquitous in m aterials and are
especially im portan t in min
erals, ceramics, semiconductors, catalysts, and polymers.
Obviously, it is very im portant
to understand and investigate the effects of quadrupolar
interactions in solid s ta te NMR.
The significance of electric quadrupole m oments for solid sta
te or chemical physics is th a t
they allow us to use the nuclei as microscopic probes for
exploring the internal electric field
gradients (EFGs). This is completely analogous w ith the use of
nuclear m agnetic moments
for probing internal magnetic fields (via the chemical shift
interaction). It is impossible
to produce any appreciable gradients by direct external means,
bu t a strong gradient can
arises from internal fields, which are produced by the electric
charge d istribution near the
nucleus. These are sensitive to subtle distortions and dynamics
of the charge distribution.
Experim ental NM R techniques can be divided into two m ain
areas depending on the
m agnitude of the quadrupolar interaction. If the interaction is
extrem ely large, nuclear
resonance experim ents can be performed at zero or very low
applied magnetic field; com
prehensive review of this field nuclear quadrupolar resonance
(NQR) is beyond the scope
of this work bu t is provided in reference [29]. On the other
hand, the alternative high field
NM R studies for which the Zeeman interaction is much larger th
an the quadrupole interac
tion will be discussed in detail. The 1957 review of quadrupolar
interaction by Cohen and
Reif has been found very helpful [23] in preparing section
4.1.1.
4 .1 .1 Q u a d r u p o le H a m ilto n ia n
Consider a nucleus subject to a electrostatic potential, V(x).
The electric charge Ze of
the nucleus is d istribu ted over the nuclear volume w ith a
density p(x). The electrostatic
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C H APTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 38
H am iltonian is expressed as an integral over the nuclear
volume,
H Q = J p (x )F (x )d 3x . (4.1)
Expanding the potential in a Taylor’s series about the center of
mass of the nucleus gives
HV = J ^ x p ( x ) | + £ ( f ( ) V + \ £ o V * + ■ ■ ■ | ,
(4-2)
where Xj (j = 1, 2, 3) stands for x, y or z, the quantities with
subscript 0 are evaluated at
x = 0 and can thus be taken out of the integral. Therefore
^ = E ( ^ ) +•••■
w ith the expectation values of:
. \ d x j j o 2 j ^ \ d x j d x k J 0
the nuclear charge j p(x)d3x — Ze , the electric dipole moment J
p(x)xjd3x = Pj , (4.4)
and the electric quadrupole moment J p (x)x jxkd3x = Q',jk .
The first term in Eq. (4.3) is simply the electrostatic energy
of a point nucleus and can be
neglected since it does not depend on the nuclear size, shape,
or orientation. The second
term for the electric dipole moment vanishes for two reasons.
Firstly, in the definition of the
electric dipole moment Xj has opposite sign in opposite octants.
Secondly, the nuclear charge
distribution has inversion sym m etry p(x) = p(—x). This sym m
etry , which is equivalent
to the probability density quantum-mechanically, arises from the
parity conservation of the
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CHAPTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 39
ground sta te wave function upon reversing the signs of all the
nuclear coordinates. From
a semi-classical point of view, during tim es where outside
fields have effect, the nucleons
are in rapid precession about the direction of the nuclear spin.
The tim e average charge
distribution thus has cylindrical symmetry. This also implies m
irror sym m etry w ith the
reflection plane passing through the center of mass,
perpendicular to the ro tation axis. Thus
the average d istribution has inversion symmetry, which also
implies a vanishing octupole
term . Therefore, Eq. (4.3) is simplified as
QjkVjk + hexadecapole term + . . . , (4-5)jk
where
d2Vv* E (4-6>
is called the electric field gradient (EFG) though strictly
speaking it is the negative of this.
Now let us estim ate the order of m agnitude of each of the
expansion term s in Eq. (4.3).
For hydrogen, the first term ZeVo ke2/ao = 4.35 x 10 18 J, or
equivalently 6.5 x 1015 Hz,
where k = 8.988 x 109 J m C ~2 is Coulomb’s constant and oo =
0.529 A is the Bohr radius
of hydrogen. The quadrupole term Q'-k{d2 V/ dxjdxQ ~ e r2(e/ag)
= eVo(rn/a o )2, where
rn ~ 10-14 m is the nuclear radius. This is of the order of 10“
8 of the electrostatic energy;
i.e., 65 MHz. And similarly, the hexdecapole term is 10~~8 times
the quadrupole term (of
the order of 0.65 Hz) and th is is so small th a t it is usually
neglected.
The quadrupole tensor Q'-k is sym m etric by definition, and
therefore it has a t m ost six
independent components. It is possible to reduce the num ber of
independent com ponents
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CH APTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 40
to five by replacing the tensor with a traceless version
Qjk = 3Q,j k - 5 j k J 2 Q i i ’ (4-7)i
where 8 j k = 1 for j = k and vanishes otherwise. The second
term on the right hand side is
independent of the nuclear orientation and can be ignored since
a nuclear orientation depen
dence is necessary in order to observe hyperfine structu re or
nuclear resonance. Therefore,
we now have
h q = \ Y ,Q * v* - (4-8)jk
It is possible to further reduce the num ber of independent com
ponents to a single term .
As m entioned above, the rapid precession creates a tim e
average distribution cylindrically
symmetric about the nuclear spin, along the direction of .X3. By
symmetry, the off-diagonal
elements of the quadrupole tensor vanish and Q \\ — Q 2 2 - But
Q n + Q 22 + Q:a = 0, hence
Q n = Q ‘22 = —Q33/ 2. As a result, the quadrupole moment tensor
is expressed in term s of
its largest principal component, Q33.
In quantum mechanical term s, the rapid precession of the
nuclear charges about the
spin direction means th a t I is a “good” quantum number; it is
thus a suitable basis. We are
interested in constant spin I since the energy required is too
high to change the nuclear spin
quantum number. Using the theorem th a t the m atrix elements of
traceless, second-rank,
sym m etric tensors are proportional [86], we have
3< m'\Qjk \m > — C < m ' \ - ( l j l k + I kI j ) - 6 j
k l2\m > , (4.9)
where C is some constant, and I2 = i f + / f + 7f ■ The constant
C can be related to the
nuclear electric quadrupole moment Q, which is the expectation
value (measured in unit of
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CHAPTER 4. ELECTRIC QUADRUPOLE INTERACTIO N
proton charge e) of Q33 in the sta te w ith m = I. T ha t
is,
41
eQ = < I\Qzz\I > • (4-10)
Q has the dimensions of area and is custom arily m easured in
barns (10 28 m 2). The constant
C = e Q / I ( 2 I — 1) (4.11)
is determ ined by comparing Eq. (4.9) (m = m ! = I ) and Eq.
(4.10). Combining Eqs. (4.8)
and (4.9), the Ham iltonian becomes
< m ' \ H Q \ m > = < m ' \ ^ ( l j l k + I k I j ) -
Sj k I 2 \ m > Vjk . (4.12)j k
The nuclear quadrupole moment Q has a classical equivalent,
e Q = J p(x)(3z2 — r 2)d3x . (4-13)
This definition highlights the fact th a t the quadrupole moment
measures the departure from
the spherical sym m etry of the nuclear charge distribution. If
Q = 0, the distribution is in
spherical sym m etry; Q > 0 if the distribution is elongated
along the spin axis like a cigar
and Q < 0 if the distribution is flattened like a
pancake.
The quadrupole interaction vanishes if the potential V arises
from a spherical or cubic
charge distribution. In this situation, the off-diagonal
elements of the field gradient tensor
are zero and the three diagonal elements are equal, Vxx — Vyy =
Vzz. Then H q in Eq.
(4.8) is proportional the the trace of Q tensor, which vanishes
by definition in Eq. (4.7).
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CH APTER 4. ELEC TRIC QUADRUPOLE IN TE R A C TIO N 42
Further, the diagonal elements of the gradient tensor also
vanish a t the center of symmetry.
The potential V satisfies the Laplace equation V 2V = Vxx + Vyy
+ Vzz = 0 a t the nuclear
center since the potential arises from charges o ther th an th a
t of the nucleus; while the three
diagonal elements are all equal, accordingly, the only solution
is th a t they are all zero. The
Laplace equation implies th a t the field gradient tensor is
also traceless.
In order to conveniently relate it to selection rules, the Ham
iltonian may be expressed
in term s of raising and lowering operators. Assisted by the
Laplace equation (V 2F = 0),
we obtain
H q = 4 / ( 2 ? - 1) [(37' " l2)Vo + {I+h + Iz I^ V- i + V - 1*
+ W Vl + J+ F- 2 + J- V*] ’
(4.14)
where
V 0 = V z z ,
V± 1 = V xz ± iVyz , (4.15)
L±2 = 2 (.Vxx ~~ Vyy) i t iVXy .
It is always possible to choose a principal axes system (PAS)
where only the diagonal
elements of the sym m etric field gradient tensor Vij are
nonzero. Moreover, the diagonal
e lem en ts sa tis fy th e L ap lace e q u a tio n , V 2F = 0, a
n d th u s o n ly tw o p a ra m e te rs a re need ed
to characterize the field gradient: the principal field gradient
component qzz and asym m etry
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CH APTER 4. ELECTRIC QUADRUPOLE IN TERA C TIO N
param eter r/ are defined as
43
CQzZ — Vzz !
r, = Vxx~ Vyy . (4.16)V zz
If we orient the principal axes so th a t the z direction has
maximum (m agnitude) gradient,
and the x direction minimum (also m agnitude), then the value of
q lies in the range 0 to 1.
The Ham iltonian in the principal axes system thus simplifies
to
h q = 470TT) K34 - A + 1(4 - 4)1 •
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CHAPTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 44
For most quadrupole isotopes, the nuclear quadrupole moments Q
are determ ined by
a variety of experim ents including, bu t not lim ited to,
atomic beam, Coulomb excitation
reorientation, Laser resonance, muonic X-ray hyperfine
structure, NQR, and NM R [98].
These values are well docum ented [98, 42]; nuclear quadrupole
moments for several nuclei
are given in Table 4.1.
T able 4.1: Quadrupole moments [98, 42].
Nucleus Q [barn]2H +0.002860(15)6 Li -0.00083(8)170 -0.0257823Na
+0.1006(20)25 Mg +0.201(3)45 Sc -0.22 (1)93Nb -0.32 (2)
However, even if Q were known exactly, the principle value of
the electric field gradient
tensor (qzz) calculated in Eq. (4.19) is not a direct m easure
of the field gradient created by
the surrounding charges; the field gradient a nucleus
experiences is amplified by deform ation
of its ionic core. The undistorted close shell core has
spherical sym m etry and thus does
not exert a field gradient a t th e nucleus. It is the r -3
dependence of the gradient (r is
the distance from a charge to the nucleus) th a t causes the
deformed core (very close to the
nucleus) to have im portant effects on the final field gradient
experienced by the nucleus
[95]. The combined effect of all electrons is described by the
Stemheimer antishielding
factor (700);
eq = eqoil - Too), (4.20)
where qo is the field gradient due to charges other th an those
of the center ion. The factor
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CHAPTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 45
is either positive (shielding) or negative (anti-shielding) and
can be large (typical values
given in Table 4.2).
The Sternheim er anti-shielding factor is difficult to calculate
w ith high accuracy. Values
also depend on the states of the same ion (free or in crystal).
Several approaches have been
applied to calculating the Sternheimer factor; discrepancies in
these results are due to both
com putation details and the initial assum ptions. The most
recent and complete work for
free ions w ith atomic num bers ranging from 2 to 94 was done in
1995 w ithin the framework
of the local density approxim ation (LDA) [54], Though not as
complete, another work has
been performed for around th irty ions in crystals as well as in
the free sta te [90].
Table 4.2: Sternheim er factors of ions.
Ion Too freea Too free6 Too crystal6Li+ 0.262 0.249 0.255o2-
0.0892 - -13.785Na+ -5 .5 9 -5 .261 -5 .452Mg2+ -3 .7 6 -3 .5 0 3
-4 .118Sc3+ -1 3 .6 -11 .388 -23.104Rb+ -5 2 .3 -47 .664
-52.781Sr2+ -4 0 .4 -38 .893 -47.828Y3+ -3 4 .8 -31 .020
-51.985Zr4+ -3 1 .1 — -Nb5+ -2 9 .5 - —49°
“ Gusev, Reznik, and Tsitrin [54].b Schmidt, Sen, Das and others
[90], except for the Nb5+entry. c This work, see section 4.1.2.
4 .1 .3 C a lc u la t io n s o f E le c tr ic F ie ld G r a d ie
n ts
We have described the experim ental approach for determ ining
the electric field gradients.
On the other hand, it is also possible to com pute the
electronic s tructu re and then calcu
late the EFG , w ith available crystal s truc tu re data. The
conventional point-charge-m odel
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CH APTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 46
approach is based on point-charge em bedded in a background
charge density th a t is broken
into uniform component [33] and undulating components [57]. The
slow convergence of
the sum over the point charges is solved by introducing an
auxiliary convergence function
th a t separates the original sum into a rapidly converging sum
in real space and a rapidly
converging sum in the reciprocal space [34]. The polarizability
of the core of the center
atom is accounted for by the Sternheim er factors.
Another more rigorous approach calculates the EFG by local
density approxim ation
(LDA) using an extended general potential Linearized Augm ented
Plane Wave M ethod
(LAPW ) [94]. This includes the core polarization effects in the
full potential calculations.
4 .1 .4 C a s e S tu d y o f P o t a s s iu m N io b a t e
The EFG a t the niobium site in potassium niobate (KNbCb)
presents itself as an interesting
case study. The crystal s tructu re of this ferroelectric m
aterial is well studied [112]. It
undergoes an orthorhom bic to rhom bohedral phase transition a t
—50 °C. NQ R study of
93Nb shows C q = 23.1 MHz and r] = 0.80 at 20 °C (orthorhom
bic), and C q = 16.0 MHz
and y — 0.0 a t — 196°C (rhom bohedral) [25]. The difference in
the asym m etry param eter
(r/) values is readily understood by considering the orthorhom
bic structu re to be obtained
from the cubic structure by elongation along the face diagonal,
and the rhom bohedral by
elongation along the body diagonal. In the orthorhom bic phase,
if the principal z-axis of
the EFG tensor is chosen to be orthogonal to the rhom bohedral
face, then inequivalent x -
and y - axes result in a non-zero asym m etry param eter r/; for
the rhom bohedral phase, if the
z-axis is chosen to be the body diagonal, then the x - and y -
axes are equivalent and result
in a zero asym m etry param eter, rj = 0 .
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CH APTER 4. ELEC TRIC QUADRUPOLE IN TERA C TIO N 47
Using conversion factors [Eq. (4.19)] and the m easured
quadrupole moment Q = —0.32
barn, the field gradient is eqzz = 2.99 x 1021 V /m 2 at 20 °C.
A simple point-charge-
model calculation yields qo = 6 x 1019 V /m 2 [25]. Since 700
has not been specifically
calculated for niobium in crystal state , taking joo = —29.5 for
the free N b5+ ion yields
eqzz = 1.8 x 1021 V /m 2 [Eq. (4.20)]. Thus calculation predicts
the same order of m agnitude
as the experim ental value, bu t w ith large error. However, the
700 values in a crystal are
very different from those of a free ion. We may estim ate 700 in
crystal from the da ta
available; substitu tion eqzz = 2.99 x 1021 V /m 2 and eqo = 6 x
1019 V /m 2 into Eq. (4.20)
gives 700 = —49 for Nb5+ in crystal. This value is very
reasonable if we compare values for
elements in the same period R b+ , Sr2+, and Y3+ (see the last
column of Table 4.2).
At the lower tem perature of —196°, the gradient is eqzz — 2.07
x 1021 V /m 2. A recent
LDA LA PW calculation for th is phase gives eqzz = 2.9 x 1021 V
/m 2 [94], the agreement is
good but could be better.
4.2 P ertu rb a tio n E xp an sion o f E nergy L evels and T
ran